82 IV ADDITIONAL PROPERTIES OF THE REAL LINE

We omit the proofs of the following two properties:
(4.1.10) lim 1/x = 0, lim 1/x = + oo, lim 1/x = - oo.

x-+Q,x<0

(4.1.11) The mappings (x, y) -> sup(x, y) and (x, y) -> inf(x, y) are continuous
in I x I.

2. MONOTONE FUNCTIONS

Let E be a nonempty subset of the extended real line R. A mapping
/ of E into 5 is called increasing (resp. strictly increasing, decreasing, strictly
decreasing) if the relation x < y (in E) implies /(x) </(j) (resp. /(x) <f(y),
/(x) ^f(y\f(x) >f(y)); a function which is either increasing or decreasing
(resp. either strictly increasing or strictly decreasing) is called monotone
(resp. strictly monotone); a strictly monotone mapping is injective. Iff
is increasing (resp. strictly increasing), —/ is decreasing (resp. strictly de-
creasing). If/, g are increasing, and f+g is defined, f+g is increasing;
if in addition / and g are both finite and one of them is strictly increasing,
then/4- g is strictly increasing.

(4.2.1) Let E be a nonempty subset of R, and a = sup E; if a $ E, then,
for any monotone mapping f of E into R, lim /(x) exists and is equal to

x-*a,xeE

sup/(x) if f is increasing, to inf/(x) if f is decreasing. (Theorem of the

xeE JceE

monotone limit.)

Suppose for instance / is increasing, and let c = sup/(x). If c = -oo,

xeE

/is constant (equal to — oo) in E and the result is trivial; if c> — oo, for
anyA < c, there is xeEsuchthati <f(x) < c;hence,forj 6 Eandx ^ y < a9
we have by assumption b <f(x) ^f(y) < c, whence our conclusion.

(4.2.2) Let I be an interval in R; any continuous injective mapping f of I
into R is strictly monotone; any continuous strictly monotone mapping f of I
into R is a homeomorphism of I onto an interval f (I).

1. Suppose/continuous and injective; let a, b be two points of I such
that a < b, and suppose for instance f(a) <f(b). Then, for a<c <b, we
must have f(a) <f(c) <f(b); for our assumptions imply f(c) ^f(b) and
(a, + oo). Given c e R, the relations x > b, y > c/b, for b > 0, imply xy > c,